Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 8.8.14580000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_8:C_2$ |
Parity: | even |
Determinant: | 1.20.4t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{5})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 15x^{6} + 60x^{4} - 90x^{2} + 45 \) . |
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 9.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 58\cdot 59 + 18\cdot 59^{2} + 24\cdot 59^{3} + 19\cdot 59^{4} + 45\cdot 59^{5} + 28\cdot 59^{6} + 42\cdot 59^{7} + 46\cdot 59^{8} +O(59^{9})\)
$r_{ 2 }$ |
$=$ |
\( 16 + 33\cdot 59 + 44\cdot 59^{2} + 57\cdot 59^{3} + 14\cdot 59^{4} + 52\cdot 59^{5} + 13\cdot 59^{6} + 54\cdot 59^{7} + 15\cdot 59^{8} +O(59^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 18 + 2\cdot 59 + 36\cdot 59^{2} + 41\cdot 59^{3} + 17\cdot 59^{4} + 31\cdot 59^{5} + 31\cdot 59^{6} + 28\cdot 59^{7} + 14\cdot 59^{8} +O(59^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 27 + 5\cdot 59 + 53\cdot 59^{2} + 6\cdot 59^{3} + 31\cdot 59^{4} + 20\cdot 59^{5} + 45\cdot 59^{6} + 24\cdot 59^{7} + 20\cdot 59^{8} +O(59^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 32 + 53\cdot 59 + 5\cdot 59^{2} + 52\cdot 59^{3} + 27\cdot 59^{4} + 38\cdot 59^{5} + 13\cdot 59^{6} + 34\cdot 59^{7} + 38\cdot 59^{8} +O(59^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 41 + 56\cdot 59 + 22\cdot 59^{2} + 17\cdot 59^{3} + 41\cdot 59^{4} + 27\cdot 59^{5} + 27\cdot 59^{6} + 30\cdot 59^{7} + 44\cdot 59^{8} +O(59^{9})\)
| $r_{ 7 }$ |
$=$ |
\( 43 + 25\cdot 59 + 14\cdot 59^{2} + 59^{3} + 44\cdot 59^{4} + 6\cdot 59^{5} + 45\cdot 59^{6} + 4\cdot 59^{7} + 43\cdot 59^{8} +O(59^{9})\)
| $r_{ 8 }$ |
$=$ |
\( 57 + 40\cdot 59^{2} + 34\cdot 59^{3} + 39\cdot 59^{4} + 13\cdot 59^{5} + 30\cdot 59^{6} + 16\cdot 59^{7} + 12\cdot 59^{8} +O(59^{9})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,8)(4,5)$ | $0$ |
$1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
$2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
$2$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
$2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $0$ |
$2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.